Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Free Probability and Random Matrices Seminar

Probability Seminar - Sang-Gyun Youn (Queen's University)

Thursday, November 29th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Sang-Gyun Youn (Queen's University)

Title: Some questions on Jones-Wenzl projections in view of quantum information theory.

Abstract:  It is very natural to study separability or PPT property of quantum states in view of quantum information theory and the need to study those properties for so-called Jones-Wenzl projections has emerged in recent years. In this talk, I am going to introduce the notions of separability, PPT property and Jones-Wenzl projections.

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Probability Seminar - Pei-Lun Tseng (Queen's University)

Thursday, October 18th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Pei-Lun Tseng (Queen's University)

Title: Right Hilbert A-modules, Part II

Abstract:  For the last talk, we gave some motivating examples and the definition of right A-module. We will begin this talk by showing some properties of A-valued pre inner product, and then we will deduce the process to construct a right Hilbert A-module. Next, we will introduce the operators on right Hilbert A-modules, and proving some properties of the corresponding adjoint operators.

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Probability Seminar - Pei-Lun Tseng (Queen's University)

Thursday, October 11th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Pei-Lun Tseng (Queen's University)

Title: Right Hilbert A-modules

Abstract:  Last week, we gave a sketch of proving the GNS construction. We will start to introduce the matrices over a C*-algebra this week, which is an application of the GNS construction. Then, we will begin the new topic: Right Hilbert A-modules. We will deduce the process to construct the inner product on a right Hilbert A-module H. As long as we have the inner product, we can consider the orthogonality on H and compare the different between the scalar case and $A$-valued case.

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Probability Seminar - Pei-Lun Tseng (Queen's University)

Thursday, October 4th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Pei-Lun Tseng (Queen's University)

Title: A-Valued Theory

Abstract:  In order to study operator-valued probability, we need some basic knowledge of C*-algebras. In this talk, I am trying to review what is a C*-algebra. and some properties of C*-algebras. The GNS construction will be covered, which is known as the follows: every C*-algebras can be represented as a algebra of bounded linear operators on a Hilbert space. This construction gives us good starting point to study matrices over a C*-algebra. If time permits, I will introduce right Hilbert A-modules.

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Probability Seminar - Jamie Mingo (Queen's University)

Thursday, September 27th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Jamie Mingo (Queen's University)

Title: Additive Convolution and Subordination

Abstract:  Last week I reviewed the basic facts of scalar free independence. This week I will continue with a new convolution of probability measures called the free additive convolution. We will use a tool from complex analysis called subordination to get our convolution. It will then lead to a discussion of conditional expectation which will be the basis of operator valued freeness.

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Probability Seminar - Jamie Mingo (Queen's University)

Thursday, September 20th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Jamie Mingo (Queen's University)

Title: I will begin by reviewing the basic facts of scalar free independence

Abstract:  This fall the seminar will be a learning seminar with lectures by (willing) participants. The theme will be operator valued freeness. This is the non-commutative version of conditional independence, now we have independence over a subalgebra. In many cases the subalgebras are n x n matrices so this is quite a general situation.

I will begin by reviewing the basic facts of scalar free independence. The seminar will follow a recent book by D. Kaliuzhnyi-Verbovetskyi and V. Vinnikov and some lecture notes of D. Jekel.

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Probability Seminar - Benson Au (Berkeley)

Wednesday, May 23rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Benson Au (Berkeley)

Title: Rigid structures in the universal enveloping traffic space

Abstract:  For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, Cébron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that extends the trace $\varphi$. The CDM construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure.

We show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a duality between classical independence and free independence.

We apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of Bryc, Dembo, and Jiang and of Mingo and Popa. This is joint work with Camille Male.

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Probability Seminar - Camille Male (Bordeaux)

Tuesday, April 10th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Camille Male (Bordeaux)

Title: An introduction to traffic independence

Abstract:  The properties of the limiting non-commutative distribution of random matrices can be usually understood thanks to the symmetry of the model, e.g. Voiculescu's asymptotic free independence occurs for random matrices invariant in law by conjugation by unitary matrices. The study of random matrices invariant in law by conjugation by permutation matrices requires an extension of free probability, which motivated the speaker to introduce in 2011 the theory of traffics. A traffic is a non-commutative random variable in a space with more structure than a general non-commutative probability space, so that the notion of traffic distribution is richer than the notion of non-commutative distribution. It comes with a notion of independence which is able to encode the different notions of non-commutative independence.

The purpose of this task is to present the motivation of this theory and to play with the notion of traffic independence.

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